DYNAMICAL SYSTEMS ARISING FROM ELLIPTIC CURVES

arXiv:math/9907014v1 [math.DS] 2 Jul 1999

DYNAMICAL SYSTEMS ARISING FROM ELLIPTIC CURVES P. D’AMBROS, G. EVEREST, R. MILES, AND T. WARD Dedicated to the memory of Professor Anzelm Iwanik Abstract. We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose entropy is the global canonical height and whose periodic points are counted asymptotically by the real division polynomial (although the archimedean component of the map is artificial). Finally, we set out a precise conjecture about the existence of elliptic dynamical systems and discuss a possible connection with mathematical physics.

1. Introduction Let F ∈ Z[x] denote a primitive polynomial with degree d, which facQ torizes as F (x) = b i (x − αi ). Then F induces a homeomorphism TF on a compact, d-dimensional group X = XF , via the companion matrix of F . The group X is an example of a solenoid whose definition is discussed in Section 2 below. The essential properties of this dynamical system TF : X → X are as follows. 1. The topological entropy h(TF ) is equal to m(F ), the Mahler measure of F (see (1) below). 2. Let Pern (F ) denote the subgroup of X consisting of elements of period n under TF , Pern (TF ) = {x ∈ X : TFn (x) = x}. If no αi is a root of Q unity, then Pern (TF ) is finite with order |Pern (TF )| = |b|n i |αin − 1|. For background, and proofs of these statements, see [7, II] and [18]. The solenoid is a generalization of the (additive) circle denoted T. Indeed, when F is monic with constant coefficient ±1, we may take X to be Td ; the resulting map is the automorphism of the torus determined by the companion 1991 Mathematics Subject Classification. 58F20, 11G07. The first author gratefully acknowledges the support of INdAM, the third of E.P.S.R.C. grant 97700813. 1

2

D’AMBROS, EVEREST, MILES, AND WARD

matrix to F . The immanence of the circle is seen in both 1 and 2 above. In 1, we may take the definition of the Mahler measure to be the logarithmic integral of |F | over the circle, (1)

m(F ) =

Z

0

1

2πit

log |F (e

)|dt = log |b| +

d X

log+ |αi |,

i=1

where the last equality follows from Jensen’s Formula (see [7, Lemma 1.9] for proof). In 2, the periodic points formula is equivalent to evaluating the n-th division polynomial of the circle on the zeros of F . That is, if Q we take φn (x) = ζ n =1 (x − ζ), we get the formula αn − 1 = φn (α), so Q |Pern (TF )| = |bn × i φn (αi )|. In [6] and [7, IV] we set out reasons for believing in the existence of elliptic dynamical systems, regarding properties 1 and 2 as the paradigm. Assuming d = 1 for example, there ought to be a dynamical system where the immanent group is a rational elliptic curve with the zero of F corresponding to the x-coordinate of a rational point on that curve. In other words, for every elliptic curve E and every point Q ∈ E(Q) we are seeking a continuous map T = TQ : X → X on some compact space X = X(E) whose dynamical data should be described by well known quantities associated to the point on the curve. We expect the entropy of T to be the global canonical height of the point Q (a well-known analogue of Mahler’s measure) and the elements of period n should be related to the elliptic n-th division polynomial evaluated at the point Q. There now follows a brief description of this paper, explaining where to look for our main conclusions. For reasons we will present in Sections 2 and 3, it is to be expected that the underlying space X should be the adelic curve. Section 2 recalls the classical definition of the solenoid and the action F induces on it. Lind and Ward [12] re-worked the classical theory in adelic terms. They showed that the topological entropy can be decomposed into a sum of local factors, each of which is the entropy of a corresponding local action. Each of these local factors can be identified as a corresponding local component of the Mahler measure. Section 3 recalls the basic theory of elliptic curves needed. In particular, the decomposition of the global canonical height into a sum of local factors. Also, we recall that the p-adic curve is isomorphic to a simpler group, on which we may expect to define dynamical systems. In Section 4, in particular the conclusion, we will construct a dynamical system where the underlying space is a p-adic elliptic curve and where the map is induced by a point on that curve. The map in question is a p-adic analogue of the well-known β-transformation. The entropy of the map is the local canonical height of the point, and the periodic points can be counted exactly. In Section 5, we will consider how to glue together these local maps to get a global dynamical system. Here we are given an

DYNAMICAL SYSTEMS ARISING FROM ELLIPTIC CURVES

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elliptic curve E defined over Q and a rational point Q ∈ E(Q). The point Q induces a dynamical system where the underlying space is the elliptic adeles and the entropy turns out to be the global canonical height of the point Q. The construction of the map at the archimedean prime is artificial since it relies upon a priori knowledge of the height of the point (although it is a curious coincidence that the map is a classical β-transformation). We hope this will bring into better focus the construction at the non-archimedean primes where the map uses no such a priori knowledge of the height. The artificiality of the map is somewhat redeemed when we go on to show that the periodic points are counted asymptotically by the real division polynomial at the point Q. This last result makes use of some non-trivial results; one from elliptic transcendence theory and the other a result about periodic points for the classical β-transformation. Finally, in Section 6, we will make some remarks about putative elliptic dynamical systems with the precise periodic point behaviour and discuss possible connections with mathematical physics. 2. The solenoid Given F (x) = bx − a, with a, b ∈ Z coprime, let X denote the subgroup of TZ defined by (2)

X = {x = (xk ) : bxk+1 = axk }.

The group TZ is compact by Tychonoff’s theorem, and X is a closed subgroup so it too is compact, an example of a (1-dimensional) solenoid. More generally, a solenoid is any compact, connected, abelian group with finite topological dimension (see [9]). The automorphism T is defined by the left shift-action (3)

T (x)k = xk+1 .

The map T has the properties 1 and 2 of Section 1 by [18] (see also [7] for a more elementary discussion). In other words, h(T ) = log max{|a|, |b|} = m(bx − a) = m(F ) (a form of Abramov’s formula). Our assumption on the zero of F not being a unit root amounts to a 6= b, and the periodic points are given by (4)

|Pern (T )| = |b|n |φn (a/b)| = |bn − an |.

At the end of this section, we will show how the periodic points formula (4) comes about. In order to motivate the name, and what follows, we will now give a second equivalent definition of the solenoid and the action of T upon it. Define X \ to be the topological dual of the ring Z[1/ab], written X = Z[1/ab]. Then

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define T to be the map which is dual to the map x 7→ ab x on Z[1/ab]. The adelic point of view arises because X is isomorphic to the quotient of Q R × p|ab Qp by the diagonally embedded discrete subgroup Z[1/ab] (this is a simple finite version of the standard adelic construction of the dual of an A-field, see [3, Section 3] or [28, Chapter IV]). Each character on R restricts 1 ˆ into X (injective ]; this induces a map from R ∼ to a character on Z[ ab =R 1 since Z[ ab ] is dense in R). The fact that the real line is ‘wrapped’ densely into the compact group X accounts for the name solenoid. The group X Q is a semi-direct product of T by p|ab Zp . The action does not preserve the various local components, but a direct calculation of the entropy formula is possible (see [27]). Lind and Ward simplified this by working with the adeles proper, which live as a covering space to the one above. In that context, the map on each component is simply multiplication by a/b. Their approach involves tensoring the dual of X with Q which gives quick access to the standard results on adeles but destroys any periodic point behaviour (see [12, Section 3]). It is probably beneficial to keep both points of view in mind. The elliptic system in Section 5 has the elliptic adeles as the base group, and for the finite primes, the local map is the local β-transformation by a/b. Thus it resembles the systems defined on both the solenoid and its adelic cover. Finally, we examine how the periodic points formula (4) comes about. This will be instructive in Section 6, when we consider a possible elliptic analogue. Suppose b = 1 so that F (x) = x − a and consider first the simpler case where the underlying space X is the (additive) circle T. The map sends x ∈ T to TF (x) = ax mod 1. Thus, the points of period n are the solutions of the equation an x = x or (an − 1)x = 0. Clearly there are |an − 1| = |φn (a)| solutions, which is the division polynomial evaluated at a. When the underlying space is the solenoid X as in (2), the map is the left shift T as in (3), so the points of x ∈ X having period n correspond to periodic vectors y of length n. The linear equation generated by such a vector is of the form Cy = 0, where C is the n × n circulant matrix on the row (a, −1, . . . ). The number of solutions y ∈ Tn of this equation, and hence the number of periodic points, is easily verified (see [7, Lemma 2.3]) to be | det(C)|. From the well-known properties of circulants, this is equal to |an − 1| = |φn (a)|.

3. Elliptic curves In this section we will recall some basic results about elliptic curves and fix the notation. A good account of elliptic curves can be found in [19] and [21]; all that follows in this section can be found in those two volumes.


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Denote by E an elliptic curve defined over a field K, and by E(K) the group of points of E having co-ordinates in K. When K = Q, Mordell’s theorem says that E(Q) is finitely generated and the torsion-free rank is ˆ : E(Q) → R the global canonreferred to simply as the rank. Denote by h ical height on E(Q), a well known analogue of Mahler’s measure (see (1)). Denote by λp the local canonical height relative to the p-adic valuation. The formula that follows gives an important decomposition of the global height as a sum of local heights (see [19, VIII] and [21, VI]): (5)

ˆ h(Q) =

X

λp (Q), for Q ∈ E(Q).

p≤∞

For finite p, whenever Q has good reduction under p, (6)

λp (Q) =

1 2

log max{|x(Q)|p , 1}.

In general, the local height is only defined up to the addition of a constant. The definition (6) agrees with the one in [20]. In [19], each local height is normalized by adding a constant to make it isomorphism-invariant. Whether normalized or not, (5) still holds. If K = R, the curve E(R) is isomorphic to either T or C2 × T (see [21, V.2]). Denote by E1 (R) the connected component of the identity, which is always isomorphic to T. If K = Qp , the curve E(Qp ) can be reduced modulo p. The set of points having non-singular reduction is denoted by E0 (Qp ) and the kernel of the reduction is denoted by E1 (Qp ). For odd ∼ primes p, there is an isomorphism E1 (Qp ) −→ pZp. This isomorphism is essentially a logarithm and it comes from the theory of formal groups. The situation when p = 2 is similar; for details, see [19, IV]. These isomorphisms are analogous to the one from E1 (R) to T. The local isomorphisms for all primes p play a very important role in the development of dynamical systems because they allow actions on the additive local curves to be transported to the local curves proper. Consider the analogous situation in Section 1, where the immanent group is the circle. When d = 1 for example, there is an isomorphism (the logarithm) from the circle to the additive group [0, 1). The action on the circle really arises from an action on [0, 1) which is then lifted via the logarithm to the circle itself. In the elliptic case, the local curve is isomorphic (via the elliptic logarithm) to an additive group. Subsequently, when we define an action on the p-adic curve, it will be one that is lifted from the additive curve. Thus, the dynamical systems which arise when the immanent group is the elliptic curve are exactly analogous to the case where the immanent group is the circle (or more generally, the solenoid). Finally, we recall the elliptic analogue of the division polynomial xn − 1 on the circle. If E denotes an elliptic curve defined over Q then without

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loss of generality it is defined by a generalized Weierstrass equation with integral coefficients. There is a polynomial ψn (x) with integer coefficients having degree n2 − 1 and leading coefficient n2 whose zeros are precisely the x co-ordinates of the points of E having order dividing n; for details see [21]. Later, we will consider the monic polynomial νn (x) of degree n − 1, whose zeros are the x co-ordinates of the non-identity points in E1 (R) having order dividing n, (7)

νn (x) =

Y

(x − x(Q)).

nQ=O O6=Q∈E1 (R)

The coefficients of νn (x) are real algebraic numbers. 4. The β-transformation and a p-adic analogue A comprehensive introduction to ergodic theory can be found in [24]. Here we just recall the definitions of ergodicity and entropy before examining in more detail the β-transformation and introducing its p-adic analogue. Let T : X → X be a measure-preserving transformation on the probability space (X, µ). Then T is ergodic if the only almost-everywhere invariant sets are trivial, in other words if µ(T −1E∆E) = 0 implies that µ(E) = 0 or 1, where ∆ is the symmetric difference. Given two open covers A, B of the compact topological space X, define their join to be A ∨ B = {A ∩ B | A ∈ A, B ∈ B}, and define the entropy of A to be H(A) = log N(A) where N(A) is the number of sets in a finite subcover with minimal cardinality. The topological entropy of a continuous map T : X → X is defined to be 



n−1 _ 1 h(T ) = sup lim H  T −j (A) , A n→∞ n j=0

where the supremum is taken over all open covers of X (see [1]; the topological entropy is a measure of orbit complexity introduced as an analogue of the measure-theoretic entropy). The β-transformation Tβ is defined for real β > 0 on the interval [0, 1) by Tβ (x) = {βx} = βx (mod 1). If β > 1, the β-transformation preserves an absolutely continuous probability measure with respect to which it is ergodic [16], the (measure–theoretic and topological) entropy is h(Tβ ) = log β (see [13] and [14], [10]) and (see [8]) the asymptotic growth rate of the periodic points equals the entropy. The result about the asymptotic growth rate will be applied in Section 5, (see (13)). Strictly speaking, the definition of topological entropy in terms of open covers does not apply to the classical β-transformation because it has a discontinuity; the topological entropy referred to is that of an associated shift system (see [24, Section 7.3]). If


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β ≤ 1, the map is simply multiplication by β. If β < 1, Tβ does not preserve an absolutely continuous measure, it has topological entropy zero and has no periodic points apart from 0. In all cases, the entropy is h(Tβ ) = log+ β. Now we define a p-adic analogue of the β-transformation. For any q ∈ Qp , define a map denoted Tq , sometimes referred to as the q-transformation, as P i follows. Let x be a generic element of Zp and write qx = ∞ i=m bi p . Define Tq (x) =

∞ X

bi pi .

i=max{0,m}

In other words, Tq multiplies by q and cuts away the fractional tail in order to come back to Zp . Note that Tq could be defined over pZp in an analogous way, and the ergodic properties would not change once the Haar measure had been normalized again. 1. If |q|p ≥ 1, the map Tq preserves Haar measure on Zp . 2. If |q|p < 1 then Tq is multiplication by q, and it only preserves the point mass at the identity. 3. Q The ring of p-adic integers Zp is homeomorphic to the space X = n∈N Y of one-sided sequences with elements in Y = {0, . . . , p − 1}, and T1/p is conjugate to the left shift σ on X. Theorem 4.1. The topological entropy of the p-adic q-transformation is given by h(Tq ) = log+ |q|p . Proof. We follow Bowen [2] and compute the topological entropy as a volume growth rate. It is a straightforward computation to check that that Haar measure on Zp is Tq -homogeneous, so (see [2, Proposition 7]) (8)

1 h(Tq ) = lim lim sup − log µ m→∞ n→∞ n

where Bm = pm Zp . If |q|p ≤ 1, Tq−1 Bm ⊃ Bm so n−1 \

n−1 \ k=0

Tq−k Bm = Bm ,

k=0

and (8) gives h(Tq ) = 0 = log+ |q|p. If |q|p = pr > 1, then Tq−1 Bm = Bm+r , so n−1 \

Tq−k Bm = Bm+rn ,

k=0

so by (8) h(Tq ) = r log p = log+ |q|p .

Tq−k Bm

!

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Theorem 4.2. Let q ∈ Qp with |q|p ≥ 1. The map Tq is ergodic with respect to Haar measure for |q|p > 1, and is not ergodic for |q|p = 1. Proof. Assume |q|p > 1 and let A denote the algebra of all finite unions of measurable rectangles and suppose E is a measurable set invariant under Tq . For any given ǫ > 0 it is possible to choose A ∈ A with µ(E △ A) < ǫ, and thus |µ(E) − µ(A)| < ǫ. Choose n such that B = Tq−n A depends upon different co-ordinates from A: then µ(A ∩ B) = µ(A)µ(B) = (µ(A))2 . Also, µ(E △ B) < ǫ, and µ(E △ (A ∩ B)) ≤ µ((E △ A) ∪ (E △ B)) < 2ǫ. Thus, |µ(E) − µ(A ∩ B)| < 2ǫ and |µ(E) − µ(E)2 | < 4ǫ. Since ǫ is arbitrary, this implies µ(E) = 0 or 1 and thus T is ergodic. When q is a unit, the open sets of the form pn Zp for n ≥ 1 are all invariant under Tq . Remark 4.3. Notice that in this setting ergodicity and mixing coincide. Coelho and Parry have studied the ergodic decomposition of Tq when q is a unit (see [4]). A consequence of properties 1 and 2 for the systems in Section 1 is that the logarithmic growth rate of the periodic points coincides with the entropy (see [7]). That this also holds for Tq follows from the next result. Theorem 4.4. Given q ∈ Qp \ U, where U denotes the set of unit roots in Qp , let Tq denote the q-transformation on Zp . Then (9)

log |Pern (Tq )| = n log+ |q|p .

Proof. Firstly, consider the case |q|p < 1. Then for n → ∞ Tqn (x) → 0 for all x ∈ Zp . Thus Tq has only one periodic point (zero) and both sides of (9) are zero. When |q| = 1, the action of q on Zp is simply multiplication, so the periodic points are solutions to the equation q n x = x. Since q is not a unit root, there are no periodic points except x = 0, so (9) holds. Finally suppose |q|p > 1. If q = p−k with k > 0, the periodic points P i are easy to determine. We have Tqn (x) = ∞ i=0 ai+nk p and the solutions to Tqn x = x are given by the pkn points with ai+nk = ai for i = 0, . . . , kn − 1. Thus, both sides of (9) are equal to nk log p. In general, suppose |q|p = pk . We claim that for each integer a with 0 ≤ a < pnk , there is a unique y ∈ Zp with Tqn (a + pnk y) = a + pnk y. This follows because the left hand side is b + q n pnk y for some b ∈ Zp , which depends only upon a, q and n. Write q n pnk = v for some p-adic unit v then the equation b + vy = a + pnk y has a unique solution for y ∈ Zp . This shows that there are at least pnk solutions of Tqn x = x. That there can be no more follows because we may take the a as above as coset representatives for Zp /pnk Zp so every element x ∈ Zp is represented by some a.


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In conclusion, given any p-adic elliptic curve E and any point Q ∈ E(Qp ), we can construct a dynamical system in the following way. The curve is locally isomorphic to the group pZp and therefore to Zp . Now let Q act via the q-transformation on the additive curve, where q = x(Q). Then transport this action to the curve proper via the logarithm. This is an exact analogue of the toral dynamical systems in Sections 1 and 2. 5. Dynamics on the Elliptic Adeles From here on, let E denote an elliptic curve defined over Q and let Q ∈ E(Q). The explicit formula (6) for the local height of Q does not hold if Q has bad reduction or if p is the prime at infinity. In particular, the local height in these cases can be negative. Since the entropy of a map is never negative, we will work with points whose local heights are guaranteed to be non-negative. Claim 5.1. There exists an n ≥ 1 for which the finite-index subgroup nE(Q) ≤ E(Q) has λp (Q) ≥ 0 for all p < ∞ and Q ∈ nE(Q). Proof. Since E1 (Q) is a subgroup of finite index in E(Q) ([19, VII]), for each bad prime p there exists an integer np such that E1 (Q) has index np Q in E(Q). Let n = bad p np , then nE(Q) ≤ E1 (Q) for all bad p. Recall now that if Q has good reduction at p (and this includes the case where Q ∈ E1 (Q)) then the local height at p is given by (6) and it follows that λp (Q) ≥ 0. Define S to be the set of bad primes together with infinity. Assume that the point Q satisfies (10)

λp (Q) > 0

for all p ∈ S.

If Q ∈ nE(Q) then Q ∈ E1 (Q) for all the bad primes. It follows from (6) that the local height is actually positive. If the rank of E(Q) is not zero then nE(Q) has finite index in E(Q) so in that case, there is a large stock of points Q which satisfy (10). At the infinite prime, this amounts to assuming that Q lies in a neighbourhood of the identity. Suppose Q ∈ E(Q) is a point for which the assumption (10) holds. Define X to be the space (11)

X=

Y

E1 (Qp ).

p≤∞

The point Q induces an action TQ : X → X in the following way: (TQ )p is the q-transformation if p is finite (where q = x(Q)) and the β-transformation if p is infinite, where log β = 2λ∞ (Q). Remember that these are actions on T and pZp , but for every p, the action can be transported to E1 (Qp ) via the isomorphisms in Section 3. The statements in the following theorem are

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analogues of statements 1 and 2 in the introduction. There we supposed that the zeros of F were not torsion points of T. The assumption that Q is not a torsion point of E is built into (10): Q is a torsion point if and only ˆ ˆ if h(Q) = 0 and (10) guarantees that h(Q) > 0. Theorem 5.2. With the definitions and assumptions above, ˆ 1. the entropy of TQ is given by h(TQ ) = 2h(Q) and 2. the asymptotic growth rate of the periodic points is given by the division polynomial νn (x) in (7): log |Pern (TQ )| ∼ log |bn νn (q)| as n → ∞. Proof. By Theorem 4.1, the entropy of each component of TQ is given by log βp , where βp = β if p = ∞ and βp = max{|x(Q)|p , 1} if p is finite. Since there are only finitely many primes for which the local dynamical systems are not isometries, Theorem 4.23 in [24] applies giving X X X ˆ log βp = 2 λp (Q) = 2h(Q). h(TQ ) = h(Tβ ) + h(Tq ) = p 1} ∪ {∞}, let X ∗ = be defined component-wise as above.

Q

p∈S ∗ (Q)

and let TQ∗


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ˆ Theorem 5.3. 1. The entropy of TQ∗ is given by h(TQ∗ ) = 2h(Q), 2. the asymptotic growth rate of the periodic points is given by the division polynomial (7): log |Pern (TQ∗ )| ∼ log |bn νn (q)|, 3. TQ∗ is ergodic. Proof. For the entropy and the periodic points, the same arguments as in Theorem 5.2 holds giving the desired result. The ergodicity is proved in Theorem 4.2 The pros and cons of our construction may be summarized as follows. Firstly, we have constructed a dynamical system whose immanent group is the adelic elliptic curve. The map is defined locally by the p-adic βtransformation on the additive curve. Secondly, the construction exhibits phenomena which resemble those in the solenoid case. Against these comments we must set the following. Firstly, the maps we are using are not continuous because of the discontinuity of the classical β-transformation. The effect upon the map TQ is to deny continuity at infinity on the archimedean component. Secondly, we would have preferred to see periodic point behaviour which was counted precisely by the usual elliptic division polynomial (rather than just asymptotically by the real division polynomial). Thirdly, the map at the archimedean prime uses a priori knowledge of the archimedean height of the point. Fourthly, we made special assumptions to guarantee that each local height was non-negative. Although these assumptions were natural, at the infinite prime and each bad prime we assumed our point was to be found in a neighbourhood of the identity, we would have preferred not to have needed any assumptions. In the next section, we will discuss how these deficiencies might be overcome. 6. Putative Elliptic Dynamics Suppose E denotes an elliptic curve defined by a generalized Weierstrass 2 equation with integral coefficients. For each n ∈ N, let ψn (x) = n2 xn −1 + ... denote the n-th division polynomial. Let Q denote a non-torsion rational point on E, with x(Q) = a/b. It is tempting to conjecture that there must a compact space X with a continuous action TQ : X → X whose entropy ˆ of is given by h(TQ ) = 2h(Q), and whose periodic points are counted, in 2 the sense of Section 1, by En (Q) = |bn −1 ψn (a/b)|. The sequence En (Q) is certainly a divisibility sequence like its toral counterpart. However, there is a problem in making the obvious conjecture. Hitherto, the systems we have considered have been Z-actions (see [7] or [18] for the definition of Zd -action). Recent work (see [15]) makes it unlikely that the sequence En (Q) counts periodic points for a Z-action. Indeed, when E is

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given by the equation y 2 + y = x3 − x and Q is the point Q = (0, 0), it follows from [15] that En (Q) cannot represent periodic point data for any Z-action. The sequence begins 1,1,1,1,2 . . . so it violates the divisibility condition (with n = 5) given in (3) of [15]. What does seem possible is that En (Q) represents periodic point data for a Z2 -action. The two reasons for saying this are firstly that the growth rate of the sequence is quadratic exponential in n (see [7, Theorem 6.18]). Thus, the sequence is more likely to represent |Per(nZ)2 (T )|, for some Z2 -action T , where (nZ)2 represents the subgroup Z2 having index n2 and consisting of all (x, y) with n|x and n|y. This would make it consistent with the known properties of algebraic Z2 actions (see [11, Theorem 7.1]). Secondly, the general feeling persists that natural maps on elliptic curves tend to be quadratic. We conjecture that for every rational elliptic curve E and every rational point Q ∈ E(Q), there is a (necessarily infinite dimensional) compact space X with a continuous Z2 -action TQ : X → X having the following properties: ˆ 1. the entropy of TQ is given by h(TQ ) = 2h(Q), and 2. the periodic points are counted by ψn in the sense that |Per(nZ)2 (TQ )| = |bn

2 −1

ψn (a/b)|.

The integral case (where b = 1) of the conjecture is already challenging. Suppose then b = 1 and we seek an action of the integral point Q, where x(Q) = a. We would hope to recognize the sequence |ψn (a)| in some natural way as counting periodic points. In [7, VI.4], we noted that the numbers |ψn (a)| arise as determinants of a nested sequence of integral n × n Hankel matrices. We suggest that these matrices might be the analogues of the circulant matrices C in Section 2. We hope that an action with such beautiful dynamical data would not be deficient in the way that the map of Section 5 was deficient. In particular, the potential negativity of the local heights need not seem such a threat. Although a negative entropy cannot exist, nonetheless, the difference between two non-negative entropies can make sense. If one dynamical systems extends another then the difference between their two entropies represents the entropy across the fibres. This raises the possibility that a phenomenon such as bad reduction might well have a dynamical interpretation. Interest in our conjecture (see [7, Question 14]) is heightened because of the connection with the following remarkable circle of ideas. On the one hand, mathematical physicists have studied the dynamics of integrable systems (see [22], [23]). Here, inter alia, one looks for meromorphic maps on the complex plane which commute with polynomials. It is a classical result of Ritt (see [17]) that all non-trivial examples arise from the exponential function or the elliptic functions associated to some lattice. Coincidentally,


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Morgan Ward (see [25], [26]) showed that all integer sequences satisfying a certain natural recurrence relation arise from the exponential function or the elliptic functions associated to some lattice, suitably evaluated. In the exponential case, these sequences can always be identified with the periodic point data for toral automorphisms. It is hoped that in the elliptic case also, the sequences |ψn (a)| represent the periodic point data for some elliptic systems. That being so, a new chapter in integrable systems could be written, yielding further inter-play between elliptic curves and mathematical physics. References [1] R. Adler, A. Konheim, and M. McAndrew. Topological entropy. Transactions of the Amer. Math. Soc., 114:309–319, 1965. [2] R. Bowen. Entropy for group endomorphisms and homogeneous spaces. Transactions of the Amer. Math. Soc., 153:401–414, 1971. [3] V. Chothi, G. Everest, and T. Ward. S-integer dynamical systems: periodic points. Journal f¨ ur die Reine und. angew. Math., 489:99–132, 1997. [4] Z. Coelho and W. Parry. Ergodic decomposition of p-adic multiplications. Preprint. [5] S. David. Minorations des formes linéaires de logarithmes elliptiques. Mem. Soc. Math. France, 62:143pp, 1995. [6] G. Everest and T. Ward. A dynamical interpretation of the global canonical height on an elliptic curve. Experimental Math., 7:305–316, 1998. [7] G. Everest and T. Ward. Heights of Polynomials and Entropy in Algebraic Dynamics. Springer, London, 1999. [8] L. Flatto, J.C. Lagarias, and B. Poonen. The zeta function of the beta transformation. Ergodic Theory and Dynamical Systems, 14:237–266, 1994. [9] E. Hewitt and K. Ross. Abstract Harmonic Analysis. Springer, New York, 1963. [10] F. Hofbauer. β–shifts have unique maximal measures. Monatshefte Math., 85:189– 198, 1978. [11] D.A. Lind, K. Schmidt, and T. Ward. Mahler measure and entropy for commuting automorphisms of compact groups. Inventiones Math., 101:593–629, 1990. [12] D.A. Lind and T. Ward. Automorphisms of solenoids and p-adic entropy. Ergodic Theory and Dynamical Systems, 8:411–419, 1988. [13] W. Parry. On the β–expansions of real numbers. Acta. Math. Acad. Sci. Hungar., 11:401–416, 1960. [14] W. Parry. Representations for real numbers. Acta. Math. Acad. Sci. Hungar., 15:95– 105, 1964. [15] Y. Puri and T. Ward. Growth of orbits: what is possible? Preprint, 1998. [16] A. Rényi. Representations for real numbers and their ergodic properties. Acta. Math. Acad. Sci. Hungar., 8:477–493, 1957. [17] J.F. Ritt. Permutable rational functions. Transactions of the Amer. Math. Soc., 25:399–448, 1923. [18] K. Schmidt. Dynamical Systems of Algebraic Origin. Birkhäuser, Basel, 1995. [19] J.H. Silverman. The Arithmetic of Elliptic Curves. Springer, New York, 1986. [20] J.H. Silverman. Computing heights on elliptic curves. Math. Comp., 51:339–358, 1988. [21] J.H. Silverman. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, New York, 1994.

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[22] A.P. Veselov. What is an integrable mapping? In V.E. Zakharov, editor, What is integrability?, pages 251–272. Springer, New York, 1991. [23] A.P. Veselov. Growth and integrability in the dynamics of mappings. Commun. Math. Phys., 145:181–193, 1992. [24] P. Walters. An Introduction to Ergodic Theory. Springer, New York, 1982. [25] M. Ward. The law of repetition of primes in an elliptic divisibility sequence. Duke Math. Journal, 15:941–946, 1948. [26] M. Ward. Memoir on elliptic divisibility sequences. Amer. Journal of Math., 70:31– 74, 1948. [27] T. Ward. The Entropy of Automorphisms of Solenoidal Groups. Master’s thesis, The University of Warwick, 1986. [28] A. Weil. Basic Number Theory. Springer, New York, third edition, 1974. School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK E-mail address: [email protected]